The shape of a memorised random walk

Abstract

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.

arXiv commentIn the journal published version, there is a misprint in the statement of Theorem 1 p.6, one should take the limit with $c\to \infty .$

References in corpus (6)

• Random walks with preferential relocations to places visited in the past and their application to biology
• Depletion-Controlled Starvation of a Diffusing Forager
• Convex Hulls of Random Walks: Large-Deviation Properties
• Mean perimeter and mean area of the convex hull over planar random walks
• Optimally Frugal Foraging
• Does Greed Help a Forager Survive?

Cited by in corpus (1)

• Size and shape of tracked Brownian bridges